Quantum Mpemba effect for operators in open systems

The quantum Mpemba effect concerns with anomalous relaxation of quantum states that evolves either under unitary or non-unitary dynamics. In the context of open quantum systems, while most studies focus on quantum states evolving under completely positive trace-presing dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, we demonstrate that an analogous effect can arise at the level of operators. In particular, we show that operators that evolves under the adjoint Liouvillian -- despite not being a trace-preserving map -- can still exhibit a genuine Mpemba effect. We derive general conditions under which this phenomenon can occur and validate our predictions for three different open quantum setups. Our results broaden the scope of the Mpemba effect in quantum systems and provide a framework for controlling the relaxation of physically relevant observables.